# Subsequential scaling limits for Liouville graph distance

## TL;DR

This paper investigates the scaling behavior of Liouville graph distance, demonstrating tightness and subsequential limits as the measure threshold approaches zero, revealing consistent diameter and typical distance scales.

## Contribution

It establishes the tightness and existence of subsequential scaling limits for Liouville graph distance as the measure threshold tends to zero.

## Key findings

- Renormalized Liouville graph distance is tight.
- Subsequential scaling limits exist as the measure threshold approaches zero.
- Diameter scales similarly to typical endpoint distances.

## Abstract

For $0<\gamma<2$ and $\delta>0$, we consider the Liouville graph distance, which is the minimal number of Euclidean balls of $\gamma$-Liouville quantum gravity measure at most $\delta$ whose union contains a continuous path between two endpoints. In this paper, we show that the renormalized distance is tight and thus has subsequential scaling limits at $\delta\to 0$. In particular, we show that for all $\delta>0$ the diameter with respect to the Liouville graph distance has the same order as the typical distance between two endpoints.